PROBABILITY  
Course: Foundations of Statistics for Data Analytics and  
Machine Learning Using Excel  
@ DV Data Analytics, Bangalore  
Instructor:  
Dr. Ujwal Deep Kadiyam  
 
Counting  
 
Counting  
Counting is a fundamental concept in combinatorics.  
Helps determine possible outcomes, optimize decisions, and solve  
real-world problems.  
Two key methods: Permutations and Combinations.  
Used in probability, statistics, Engineering and computer science.  
The Counting Principle  
It states that if there are m ways to perform one action and n ways to  
perform another, then there are m × n ways to perform both actions.  
This principle is used to calculate the total number of possibilities  
when multiple events or choices are involved.  
Extends to multiple events: m1 × m2 × · · · × mk.  
Problem: A restaurant offers 3 choices of starters and 4 choices of main  
courses. How many different meal combinations can be made?  
Each appetizer choice can be paired with any main course choice.  
Using the multiplication principle there are 12(3 × 4) different meal  
combinations.  
 
Example: Counting Principle (Multiplication Principle)  
Problem: How many different passwords can be created if a password  
consists of 3 letters (A-Z) followed by 2 digits (0-9)?  
Each letter has 26 choices.  
Each digit has 10 choices.  
Since each choice is independent, we multiply:  
26 × 26 × 26 × 10 × 10 = 263 × 102  
= 17, 576 × 100 = 1, 757, 600  
Answer: There are 1,757,600 possible passwords.  
 
Rules for Counting in Sets  
If event A can occur in m ways and event B (which is independent of A)  
can occur in n ways, then the number of ways either event can occur is:  
m + n  
Example: A student should select one project and if that student can  
choose a project from two different categories:  
3 AI projects  
2 Robotics projects  
Since the student cannot choose from both categories at the same time,  
the total number of choices is:  
3 + 2 = 5  
 
Limitations of addition and Multiplication rule:  
Choosing a Committee with Restrictions  
A school is forming a 2-member committee from a pool of 4 students:  
A, B, C, D  
Using Multiplication rule, we will get 4 × 3 = 12 different committees.  
The following are the committees AB, AC, AD, BC, BD, CD.  
Multiplication Rule Fails.  
 
Incorrect Counting Example  
Three officers—a president, a treasurer, and a secretary—are to be chosen  
from four people: Ann, Bob, Cyd, and Dan.  
Ann cannot be president.  
Either Cyd or Dan must be secretary.  
A mistaken approach:  
1
Choose a president from {Bob, Cyd, Dan} → 3 choices.  
2
Choose a treasurer from the remaining 3 people → 3 choices.  
3
Choose a secretary from {Cyd, Dan} → 2 choices.  
Applying the multiplication principle incorrectly:  
3 × 3 × 2 = 18  
This answer is incorrect because choosing a treasurer should be from only  
2 remaining candidates.  
 
Officer Selection  
Secretary: Cyd  
Treasurer:Ann  
Secretary: Cyd  
Bob  
Dan  
Secretary: Cyd  
Secretary: Dan  
Treasurer:Ann  
Secretary: Dan  
President  
Treasurer:Ann  
Bob  
Cyd  
Treasurer:Cyd  
Secretary: Dan  
Treasurer:Dan  
Secretary: Cyd  
Treasurer:Bob  
Secretary: Dan  
 
Permutations  
 
Permutations  
Definition: The number of ways to arrange r objects from n distinct  
objects.  
n!  
nPr = P(n, r) =  
(n − r)!  
Order matters.  
Example: Arranging 2 books from a set of 4.  
All permutations of A,B,C,D:  
AB  
AC  
AD  
BA  
BC  
BD  
CA  
CB  
CD  
DA  
DB  
DC  
5!  
3! · 2!  
120  
P(5, 3) =  
=
= 10  
6 · 2  
 
Permutations with Repetition  
Definition: The number of ways to arrange r objects from n objects when  
repetition is allowed.  
Prep(n, r) = nr  
Order matters.  
Objects can be repeated.  
Example: Creating a 4-digit PIN using 10 digits (0-9):  
Prep(10, 4) = 104 = 10, 000  
 
Combinations  
 
Combinations  
Definition: The number of ways to choose r objects from n distinct  
objects without considering order.  
ꢀ ꢁ  
n
r
n!  
nCr = C(n, r) =  
=
r!(n − r)!  
Order does not matter.  
Example: Selecting 3 students from a group of 5.  
All possible selections:  
ABC  
ABD  
ABE  
ACD  
ACE  
ADE  
BCD  
BCE  
BDE  
CDE  
5!  
120  
C(5, 3) =  
=
= 10  
3! · 2!  
6 · 2  
 
Combinations with Repetition  
Definition: The number of ways to choose r objects from n distinct  
objects when repetition is allowed.  
ꢀ
ꢁ
n + r − 1  
(n + r − 1)!  
Crep(n, r) =  
=
r
r!(n − 1)!  
Order does not matter.  
Objects can be repeated.  
Example: Choosing 3 scoops of ice cream from 5 flavors (where  
repeats are allowed):  
ꢀ
ꢁ
ꢀ ꢁ  
5 + 3 − 1  
7
Crep(5, 3) =  
=
= 35  
3
3
 
Key Differences  
Permutations consider order; combinations do not.  
P(n, r) > C(n, r) for r > 1.  
Use permutations when arranging, and combinations when selecting.  
 
Example: Counting  
Problem: A password consists of 4 distinct letters chosen from the  
English alphabet (26 letters). How many such passwords are possible?  
 
Example: Counting  
Problem: A password consists of 4 distinct letters chosen from the  
English alphabet (26 letters). How many such passwords are possible?  
Since order matters, we use permutations:  
26!  
26!  
22!  
P(26, 4) =  
=
(26 − 4)!  
Expanding the factorial:  
26 × 25 × 24 × 23 = 358800  
Thus, there are 358,800 possible passwords.  
 
Summary Table with Excel Functions  
Formula Without Repetition  
With Repetition  
n!  
Permutations P(n, r)  
(n−r)!  
nr  
Excel Formula  
PERMUT(n, r)  
PERMUTATIONA(n, r)  
(n+r−1)!  
n!  
Combinations C(n, r)  
Excel Formula  
r!(n−r)!  
COMBIN(n, r)  
r!(n−1)!  
COMBINA(n, r)  
 
Excel Functions for Counting and Probability  
Function  
RAND()  
Description  
Example  
=RAND()  
Output  
0.423 (varies)  
Generates a random number between 0 and 1  
Generates a random integer between two bounds  
Returns the factorial of a number  
RANDBETWEEN  
FACT  
=RANDBETWEEN(1, 6)  
=FACT(5)  
4 (varies)  
120  
PERMUT  
Returns the number of permutations (order matters)  
Returns the number of permutations allowing repetitions  
Returns the number of combinations (order does not matter)  
Returns the number of combinations allowing repetitions  
Rounds a number to a specified number of digits  
=PERMUT(5, 3)  
=PERMUTATIONA(5, 3)  
=COMBIN(5, 3)  
=COMBINA(5, 3)  
=ROUND(0.123456, 2)  
60  
PERMUTATIONA  
COMBIN  
125  
10  
COMBINA  
ROUND  
35  
0.12  
 
Probability  
 
Introduction to Probability  
Probability of an event is a measure (number) of the chance with  
which we can expect the event to occur.  
It is quantified as a number between 0 and 1.  
Defined as:  
Number of favorable outcomes  
P(A) =  
Total number of outcomes  
Probability values range from 0 (impossible event) to 1 (certain  
event).  
 
Introduction to Probability  
Basic Concepts:  
Outcome: A single possible result of a random experiment. It is an  
element of the sample space S.  
Sample Space (S): The set of all possible outcomes.  
Event (E): A subset of the sample space.  
|E|  
Probability of an event: P(E) =  
Example: Rolling a Die  
(for equally likely outcomes).  
|S|  
Sample space: S = {1, 2, 3, 4, 5, 6}.  
Outcome: Rolling a 3 (i.e., outcome = 3).  
Event A: Rolling an even number, A = {2, 4, 6}.  
Key Difference:  
An outcome is a single result.  
An event consists of multiple possible outcomes.  
 
Outcome, Sample Space and Event  
 
Types of Probability: Classical  
Classical Probability (Theoretical Probability)  
Based on equally likely outcomes.  
Formula: P(E) = MN , where M is the number of favorable outcomes  
and N is the total outcomes.  
Example: Rolling a fair six-sided die. The probability of rolling a 4:  
1
P(4) =  
6
 
Types of Probability: Empirical  
Empirical Probability (Experimental Probability)  
Based on observed data from experiments.  
Number of times E occurs  
Total number of trials  
Formula: P(E) =  
.
Example: If a coin is flipped 100 times and heads appear 55 times:  
55  
P(H) =  
= 0.55  
100  
 
Types of Probability: Subjective  
Subjective Probability  
Based on intuition, personal judgment, or expert opinions.  
Used when no historical data is available.  
Example: A doctor estimates a 70% chance of recovery for a patient  
based on symptoms and experience.  
 
When Does an Event Occur?  
Question: If one outcome in an event happens, does the event occur?  
Answer: Yes! If at least one outcome from the event occurs, then the  
event has occurred.  
Example: Rolling a Die  
Event A: Rolling an even number, A = {2, 4, 6}.  
If the outcome is 4, then since 4 ∈ A, event A occurs.  
If the outcome is 3, then since 3 ∈/ A, event A does not occur.  
Conclusion: An event occurs if any of its outcomes happens.  
 
Why Permutations and Combinations in Probability?  
Probability is about comparing:  
Number of favorable outcomes  
Probability =  
Total number of possible outcomes  
To find these counts, we often need:  
Permutations – when order matters  
Combinations – when order doesn’t matter  
 
Example 1: Permutation (Order Matters)  
Problem: A password consists of 3 different digits from 0–9. What is the  
probability that you guess it correctly on the first try?  
Solution:  
Total outcomes = Number of ways to arrange 3 digits from 10:  
10!  
10P3 =  
= 720  
(10 − 3)!  
Favorable outcomes = 1 (only one correct password)  
1
Probability =  
720  
 
Example 2: Combination (Order Doesn’t Matter)  
Problem: From a group of 8 students, a committee of 3 is selected. What  
is the probability that a specific group of 3 students is chosen?  
Solution:  
Total combinations:  
ꢀ ꢁ  
8
3
8!  
=
= 56  
3! · 5!  
Favorable outcomes = 1 (only one specific group)  
1
Probability =  
56  
 
Axioms and Rules of Probability  
 
Axioms of Probability  
Given a sample space S and an event E, probability satisfies:  
1
Non-negativity: P(E) ≥ 0 for all events E.  
Normalization: P(S) = 1 (Sum of probabilities of all events = 1).  
2
3
Additivity: If E1, E2, . . . are disjoint events, then  
P(E1 ∪ E2 ∪ . . . ) = P(E1) + P(E2) + . . .  
Example: Rolling a fair die  
S = {1, 2, 3, 4, 5, 6}  
1
6
P(rolling a 2) =  
3
6
1
2
P(rolling an even number) = P(2) + P(4) + P(6) =  
=
 
Basic Probability Rules  
Addition Rule: If events A and B are mutually exclusive:  
P(A ∪ B) = P(A) + P(B)  
Multiplication Rule: If events A and B are independent:  
P(A ∩ B) = P(A) × P(B)  
Complement Rule: The probability of an event not occurring:  
P(Ac) = 1 − P(A)  
 
Basic Probability Rules  
Example: Rolling a die — what’s the probability of getting a 2 or a  
5?  
 
Basic Probability Rules  
Example: Rolling a die — what’s the probability of getting a 2 or a  
5? Since 2 and 5 can’t happen at the same time:  
1
6
1
6
2
6
P(2 or 5) =  
+
=
Example: Tossing a coin and rolling a die. Probability of getting  
heads and a 4:  
 
Basic Probability Rules  
Example: Rolling a die — what’s the probability of getting a 2 or a  
5? Since 2 and 5 can’t happen at the same time:  
1
6
1
6
2
6
P(2 or 5) =  
+
=
Example: Tossing a coin and rolling a die. Probability of getting  
heads and a 4:  
1
2
1
6
1
P(H and 4) =  
×
=
12  
Example: Probability of not getting a 6 when rolling a die:  
 
Basic Probability Rules  
Example: Rolling a die — what’s the probability of getting a 2 or a  
5? Since 2 and 5 can’t happen at the same time:  
1
6
1
6
2
6
P(2 or 5) =  
+
=
Example: Tossing a coin and rolling a die. Probability of getting  
heads and a 4:  
1
2
1
6
1
P(H and 4) =  
×
=
12  
Example: Probability of not getting a 6 when rolling a die:  
1
6
5
6
P(not 6) = 1 −  
=
 
Union of sets: Addition Rule  
 
Intersection of sets: Multiplication Rule  
 
Intersection of sets  
 
Sample Space and Event  
• Event is a subset of Sample Space.  
 
Mutually Exclusive Events  
 
Mutually Exclusive Events  
Definition: Two events A and B are said to be mutually exclusive if  
they cannot occur at the same time.  
Key Characteristics:  
If one event happens, the other cannot.  
Their intersection is empty: A ∩ B = ∅.  
The probability of both occurring together is zero.  
P(A ∩ B) = 0.  
Example: Rolling a Die  
Let A be the event of rolling an even number: A = {2, 4, 6}.  
Let B be the event of rolling an odd number: B = {1, 3, 5}.  
Since no number can be both even and odd, A and B are mutually  
exclusive.  
Mathematically, P(A ∩ B) = 0.  
 
Sample Space and Mutually Exclusive Events  
• Events that cant occur at the same time.  
 
Conditional Probabilty  
 
Introduction to Conditional Probability  
Definition: The conditional probability of an event A given that event B  
has occurred is defined as:  
P(A ∩ B)  
P(A | B) =  
,
P(B) > 0.  
P(B)  
Key Points:  
Measures the likelihood of A occurring given that B has occurred.  
The probability space is reduced to B.  
Requires that P(B) > 0.  
 
Example Problem  
Example: A box contains 5 red balls and 7 blue balls. If a ball is drawn at  
random, what is the probability that it is red given that it is not blue?  
Solution:  
Let A be the event that the ball drawn is red.  
Let B be the event that the ball drawn is not blue.  
Since the only non-blue balls are red, we have P(A ∩ B) = P(A).  
Using the conditional probability formula:  
P(A ∩ B)  
5/12  
P(A | B) =  
=
= 1.  
P(B)  
5/12  
 
Example Problem  
Example: A bag contains 4 green, 3 red, and 5 yellow marbles. If a  
marble is randomly selected, what is the probability that it is red given  
that it is not yellow?  
Solution:  
Let A be the event that the marble drawn is red.  
Let B be the event that the marble drawn is not yellow.  
The total number of marbles is 4 + 3 + 5 = 12.  
The number of non-yellow marbles is 4 + 3 = 7.  
The number of red marbles is 3.  
Using the conditional probability formula:  
P(A ∩ B)  
3/12  
3
7
P(A | B) =  
=
=
.
P(B)  
7/12  
 
Example Problem  
Example: Suppose we have a deck of 52 playing cards. What is the  
probability that a card drawn is an Ace given that it is a Spade?  
Solution:  
Let A be the event that the card drawn is an Ace.  
Let B be the event that the card drawn is a Spade.  
There are 13 Spades and only 1 Ace of Spades.  
Using the conditional probability formula:  
P(A ∩ B)  
1/52  
1
P(A | B) =  
=
=
.
P(B)  
13/52  
13  
 
Independence  
 
Independent Events  
Definition: Two events A and B are said to be independent if the  
occurrence of one does not affect the probability of the other.  
Mathematically,  
P(A ∩ B) = P(A)P(B).  
Equivalent Conditions:  
P(A | B) = P(A), meaning knowing that B occurred does not  
change the probability of A.  
P(B | A) = P(B), meaning knowing that A occurred does not  
change the probability of B.  
Example: Suppose we roll a fair die and flip a fair coin. Define:  
A: Getting a 6 on the die.  
B: Getting heads on the coin.  
Since rolling the die does not affect the coin flip,  
ꢀ ꢁ ꢀ ꢁ  
1
6
1
2
1
P(A ∩ B) =  
×
=
,
12  
which matches P(A)P(B), proving independence.  
 
Examples and Misconceptions  
Example: Drawing Cards from a Deck  
Let A be the event of drawing a red card.  
Let B be the event of drawing a spade.  
Since a card cannot be both red and a spade, P(A ∩ B) = 0.  
ꢂ
ꢃ
26  
52  
13  
52  
However, P(A)P(B) =  
×
̸= 0.  
Since P(A ∩ B) ̸= P(A)P(B), A and B are not independent.  
Common Misconceptions:  
Independence is not the same as mutual exclusivity: If A and B  
are mutually exclusive, then P(A ∩ B) = 0, so they cannot be  
independent unless P(A) = 0 or P(B) = 0.  
Independence is not about equal probabilities: Events can be  
independent even if they have different probabilities.  
 
Understanding P(A ∩ B): Intersection of Events  
Case  
Formula and Explanation  
Independent Events  
P(A ∩ B) = P(A) · P(B)  
A and B do not affect each other  
(e.g., coin toss and die roll)  
Dependent Events  
P(A ∩ B) = P(A) · P(B|A)  
Probability of B depends on A happening  
(e.g., drawing 2 cards without replacement)  
Mutually Exclusive Events P(A ∩ B) = 0  
A and B cannot happen at the same time  
(e.g., getting 2 and 5 on a single die roll)  
 
Law of Total Probability  
 
Law of Total Probability  
The law of total probability is a fundamental rule in probability  
theory.  
The law of total probability provides a way to compute the probability  
of an event by considering all possible ways that event can occur.  
It provides a way to compute the probability of an event by  
considering partitioning of the sample space.  
Useful when dealing with conditional probabilities.  
If B1, B2, . . . , Bn form a partition of the sample space and P(Bi) > 0  
for all i, then for any event A:  
n
X
P(A) =  
P(A | Bi)P(Bi)  
i=1  
 
Illustration  
Consider a sample space partitioned into disjoint events B1, B2, B3 . . . B8:  
Each Bi represents a different way an event can happen.  
The probability of A is the sum of its conditional probabilities  
weighted by the probability of each Bi.  
 
Example 1: Defective Items  
A factory has two machines:  
Machine 1 produces 60% of items, and Machine 2 produces 40%.  
Machine 1 has a 5% defect rate, while Machine 2 has a 10% defect  
rate.  
What is the probability that a randomly chosen item is defective?  
 
Solution  
Define events:  
D: Item is defective  
B1: Item from Machine 1, P(B1) = 0.6  
B2: Item from Machine 2, P(B2) = 0.4  
Given:  
P(D | B1) = 0.05, P(D | B2) = 0.10  
Using the law of total probability:  
P(D) = P(D | B1)P(B1) + P(D | B2)P(B2)  
P(D) = (0.05)(0.6) + (0.10)(0.4) = 0.03 + 0.04 = 0.07  
Thus, the probability of selecting a defective item is 0.07 (or 7%).  
 
Bayes Rule  
 
Bayes Theorem  
Bayes Theorem is a fundamental result in probability theory.  
It describes how to update our belief about an event based on new  
evidence.  
Commonly used in medical diagnosis, spam filtering, and machine  
learning.  
 
Derivation of Bayes’ Theorem:  
Consider a sample space partitioned into two mutually exclusive  
events B1 and B2, such that B1 ∪ B2 = S.  
From the definition of conditional probability:  
P(A ∩ B1)  
P(A | B1) =  
P(B1)  
P(A ∩ B1)  
P(B1 | A) =  
P(A)  
Rearranging both equations:  
P(A ∩ B1) = P(A | B1)P(B1)  
P(A ∩ B1) = P(B1 | A)P(A)  
Equating the two expressions for P(A ∩ B1):  
P(A | B1)P(B1) = P(B1 | A)P(A)  
 
Derivation  
Solving for P(B1 | A) gives:  
P(A | B1)P(B1)  
P(B1 | A) =  
This is Bayes’ Theorem.  
P(A)  
Using the Law of Total Probability:  
P(A) = P(A | B1)P(B1) + P(A | B2)P(B2)  
 
Theorem Statement  
Bayes’ Theorem: Given an event A and a partition of the sample space  
B1, B2, . . . , Bn:  
P(A | Bi)P(Bi)  
P(Bi | A) =  
P(A)  
where  
n
X
P(A) =  
P(A | Bi)P(Bi)  
i=1  
LTP: If B1, B2, ..., Bn form a partition of the sample space, then for any  
event A:  
n
X
P(A) =  
P(A | Bi)P(Bi)  
i=1  
This rule helps compute the probability of an event by considering all  
possible ways it can occur.  
It is used as the denominator in Bayes’ Theorem.  
 
Example 1: Spam Filtering  
A spam filter detects spam emails based on keywords.  
Suppose 20% of emails are spam.  
Given that an email contains the word ”free”, how likely is it to be  
spam?  
Let:  
P(S) = 0.20, P(Sc) = 0.80  
P(F | S) = 0.50, P(F | Sc) = 0.05  
Using Bayes’ Theorem:  
P(F | S)P(S)  
P(S | F) =  
P(F)  
P(F) = (0.50 × 0.20) + (0.05 × 0.80) = 0.14  
0.50 × 0.20  
P(S | F) =  
= 0.714  
0.14  
The probability that an email is spam given that it contains ”free” is  
71.4%.  
 
Example 2: Medical Testing  
A disease affects 1% of a population.  
A test detects the disease with 95% sensitivity and 90% specificity.  
What is the probability that a person who tests positive actually has  
the disease?  
Using Bayes’ Theorem:  
P(T+ | D)P(D)  
P(D | T+) =  
P(T+)  
P(T+ | D) = 0.95, P(D) = 0.01  
P(T+ | Dc) = 1 − 0.90 = 0.10, P(Dc) = 0.99  
P(T+) = (0.95 × 0.01) + (0.10 × 0.99) = 0.1045  
0.95 × 0.01  
P(D | T+) =  
≈ 0.091  
0.1045  
Only 9.1% of those who test positive actually have the disease.  
 
Bayes’ Theorem and its Relationship with LTP  
LTP computes the total probability of an event.  
Bayes’ Theorem inverts this relationship to find the probability of  
causes given evidence.  
It allows us to update beliefs based on new information.  
 
Law of Total Probability vs Bayes’ Rule  
Big Picture: How They Differ  
Pizza Delivery Example  
60% orders from A (20% late), 40% from B (50% late)  
Law of Total Probability:  
P(Late) = P(Late | A) · P(A) + P(Late|B) · P(B)  
→ What’s the chance your pizza is late overall?  
Bayes’ Rule:  
P(Late|B) · P(B)  
P(B | Late) =  
P(Late)  
→ Given that it’s late, what’s the chance it came from B?  
 
Summary  
Law of Total Probability  
Think of it as...  
Looking forward from causes to outcomes  
What it helps with Calculates overall probability of an outcome from multiple scenarios  
Bayes’ Rule  
Think of it as...  
Looking backward from outcomes to causes  
What it helps with Updates the probability of a cause given that an outcome has occurred  
 
Random Variables  
 
Introduction to Random Variables  
A random variable is a numerical description of the outcome of a  
statistical experiment. It is a way to assign numbers to outcomes of a  
random experiment  
It is a function defined on a sample space, S, that associates a real  
number with each outcome in S.  
e1  
Sample space S  
R, the real line  
f(e1)  
Two main types:  
Discrete Random Variables: Take countable values (e.g., number of  
heads in a coin toss).  
Continuous Random Variables: Take an infinite number of values  
within a range (e.g., height of individuals).  
If each event is associated with its probability in random variable, it is  
called as Probability distribution.  
 
Example: Coin Toss  
Experiment: Toss a coin  
Let X be a random variable such that:  
X = 1 if Heads  
X = 0 if Tails  
X is a discrete random variable.  
 
What is a Distribution?  
A distribution tells us the probability of each value of a random  
variable.  
Shows how likely each outcome is.  
A distribution describes how values of a random variable are spread.  
It helps in understanding the likelihood of different outcomes.  
Examples: Normal, Uniform, Binomial, Poisson, etc.  
Can be a:  
Probability distribution (for discrete variables)  
Probability density function (for continuous variables)  
Creating a Distribution:  
Consider rolling a six-sided die multiple times.  
We record the outcomes and plot their frequency.  
 
Coin Toss: Distribution Example  
Outcome Value (X) Probability  
Heads  
Tails  
1
0
0.5  
0.5  
This is the probability distribution of X.  
 
Dice Roll Example  
Experiment: Roll a fair 6-sided die  
Random variable X = number on the die  
Value of X Probability  
1
2
3
4
5
6
1/6  
1/6  
1/6  
1/6  
1/6  
1/6  
This is a uniform distribution.  
 
Summary  
Concept  
Explanation  
Random Variable  
A variable that assigns numbers to outcomes  
of a random process  
Distribution  
Describes how probabilities are assigned to  
each value of the random variable  
 
Example  
Let X and Y be the outcomes of two fair 6-sided dice.  
Define a new random variable:  
D = X − Y  
Possible values of the random variable D are:  
{−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5}  
 
The Distribution  
 
Tabulated Distribution (Example)  
D
-5  
-4  
-3  
-2  
-1  
0
1
2
3
4
Probability Cumulative  
0.0278  
0.0556  
0.0833  
0.1111  
0.1389  
0.1667  
0.1389  
0.1111  
0.0833  
0.0556  
0.0278  
0.0278  
0.0834  
0.1667  
0.2778  
0.4167  
0.5834  
0.7223  
0.8334  
0.9167  
0.9723  
1.0000  
5
• Observe the Cumulative Probability  
 
Motivating a Formal Function  
Question: How do we represent a distribution mathematically?  
Answer: Using a function that tells us the probability of each value:  
For discrete random variables: Probability Mass Function (PMF)  
For continuous random variables: Probability Density Function  
(PDF)  
 
PMF: Probability Mass Function  
Definition: For a discrete random variable X, the PMF is defined as:  
p(x) = P(X = x)  
Example: Let D = X − Y be the difference of two dice rolls. Then:  
p(0) = 0.1667, p(1) = 0.1389, . . .  
Properties:  
0 ≤ p(x) ≤ 1  
P
x p(x) = 1  
 
PDF: Probability Density Function  
Definition: For a continuous random variable X, the PDF is a function  
f(x) such that:  
Z
b
P(a ≤ X ≤ b) =  
f(x) dx  
a
Properties:  
f(x) ≥ 0  
R ∞  
−∞ f(x) dx = 1  
It means The total area under the PDF curve is always 1  
Note: f(x) is not a probability — only areas under the curve give  
probabilities.  
 
PMF vs PDF  
PMF: Used for discrete variables (e.g., dice outcomes)  
PDF: Used for continuous variables (e.g., height, weight)  
Both describe how probability is distributed over outcomes  
 
Why CDF?  
Question: What is the probability that a variable is less than or equal to a  
value?  
Answer: We use the Cumulative Distribution Function (CDF) to  
capture that.  
 
Cumulative Distribution Function (CDF)  
Definition: For a random variable X, the CDF is defined as:  
F(x) = P(X ≤ x)  
P
Discrete: F(x) = t≤x p(t)  
R
x
Continuous: F(x) = −∞ f(t) dt  
 
Properties of CDF  
F(x) is non-decreasing  
limx→−∞ F(x) = 0  
limx→∞ F(x) = 1  
For continuous X, F′(x) = f(x)  
 
Example: CDF of D = X − Y  
We already saw the PMF for D.  
The CDF at a point is the sum of all probabilities up to that point.  
Example:  
F(0) = P(D ≤ 0) = p(−5) + p(−4) + · · · + p(0) = 0.5834  
 
Cumulative Distribution Function (CDF)  
The Cumulative Distribution Function (CDF) gives the probability  
that a random variable X is less than or equal to a certain value x.  
It is defined as:  
F(x) = P(X ≤ x)  
For a discrete random variable:  
X
F(x) =  
P(X = t)  
t≤x  
For a continuous random variable:  
Z
x
F(x) = −∞ f(t)dt  
The CDF is a non-decreasing function that ranges from 0 to 1.  
Example: For a standard normal variable, the CDF is given by:  
Z
x
1
2
√
F(x) =  
−∞ e−t /2dt  
2π  
 
PMF and CDF  
 
PDF and CDF  
 
Comparison of PMF, PDF, and CDF  
Probability Mass Function (PMF):  
Used for discrete random variables.  
Assigns probabilities to specific values.  
Probability Density Function (PDF):  
Used for continuous random variables.  
Represents density, not direct probabilities.  
The area under the curve represents probability.  
Cumulative Distribution Function (CDF):  
Used for both discrete and continuous random variables.  
Gives cumulative probability up to a given value.  
Always non-decreasing and ranges from 0 to 1.  
 
Summary of Applications  
To summarize, here are the main uses of these functions:  
PMF: Used for discrete random variables to calculate probabilities for  
specific outcomes.  
CMF: Useful for calculating the cumulative probability in discrete  
distributions.  
PDF: Used for continuous random variables to model probability  
densities and calculate probabilities over intervals.  
CDF: Essential for continuous variables to calculate the cumulative  
probability up to a given value.  
These functions are widely applied in fields like:  
Statistics and data analysis.  
Risk management and decision-making.  
Machine learning algorithms (e.g., Gaussian Naive Bayes classifier).  
Quality control and reliability engineering.  
 
Example: Rolling a Die  
Consider rolling a fair six-sided die.  
The outcome X is a discrete random variable with possible values:  
{1, 2, 3, 4, 5, 6}.  
Probability Mass Function (PMF):  
1
P(X = x) =  
,
x = 1, 2, 3, 4, 5, 6  
6
Expected Value:  
X
1 + 2 + 3 + 4 + 5 + 6  
6
E[X] =  
xP(X = x) =  
= 3.5  
 
Random Variables  
A random variable is a function that assigns a numerical value to  
each outcome in a sample space.  
Can be:  
Discrete: Countable outcomes (e.g., number of heads in coin tosses).  
Continuous: Uncountable outcomes (e.g., height of students).  
 
Joint Distribution: Definition  
Definton  
The joint distribution of two random variables X and Y gives the  
probability that X takes a specific value and Y takes a specific value at  
the same time.  
Notation: P(X = x, Y = y)  
 
Joint Probability Table (Discrete Case)  
Example: Joint Distribution Table  
X\Y  
1
2
3
1
2
3
0.1 0.1 0.1  
0.2 0.1 0.1  
0.05 0.1 0.15  
Each cell gives P(X = x, Y = y).  
 
Marginal Distribution: Definition  
Definton  
The marginal distribution of a random variable is the probability  
distribution of that variable alone, obtained by summing (or integrating)  
the joint distribution over the other variable.  
For discrete variables:  
X
P(X = x) =  
P(X = x, Y = y)  
y
X
P(Y = y) =  
P(X = x, Y = y)  
x
 
Finding Marginal Distributions (Example)  
From the previous table:  
Marginal of X:  
P(X = 1) = 0.1 + 0.1 + 0.1 = 0.3  
P(X = 2) = 0.2 + 0.1 + 0.1 = 0.4  
P(X = 3) = 0.05 + 0.1 + 0.15 = 0.3  
Marginal of Y is found similarly by adding down columns.  
 
Joint and Marginal Densities (Continuous Case)  
Joint Probability Density Function (pdf): f(x, y)  
f(x, y) satisfies:  
Z
Z
∞
∞
−∞ f(x, y) dx dy = 1  
−∞  
Probability that (X, Y ) lies in a region A:  
ZZ  
P((X, Y ) ∈ A) =  
f(x, y) dx dy  
A
Marginal Densities:  
Z
∞
fX(x) = −∞ f(x, y) dy  
Z
∞
fY (y) = −∞ f(x, y) dx  
 
Visualizing Joint Distributions  
Joint distributions can be visualized as 3D surfaces or contour plots.  
 
Key Points  
Joint Distribution shows relationship between two random variables.  
Marginal Distribution looks at one variable alone.  
In the discrete case, sum probabilities; in the continuous case,  
integrate.  
Helps in finding conditional distributions and checking independence.  
 
Quick Check!  
Question:  
Given the joint table below:  
X\Y  
0
1
0
1
0.3 0.2  
0.1 0.4  
Find:  
P(X = 0)  
P(Y = 1)  
 
Solution  
Marginal for X = 0:  
P(X = 0) = 0.3 + 0.2 = 0.5  
Marginal for Y = 1:  
P(Y = 1) = 0.2 + 0.4 = 0.6  
 
Independence of Random Variables  
Two random variables X and Y are independent if and only if:  
P(X = x, Y = y) = P(X = x) × P(Y = y) for all x, y  
Otherwise, X and Y are said to be dependent.  
 
Idea Behind Checking Independence  
Start from the joint probability P(X = x, Y = y).  
Compute the marginal probabilities P(X = x) and P(Y = y).  
Check if:  
P(X = x, Y = y) = P(X = x) × P(Y = y)  
for all possible pairs (x, y).  
 
Example: Joint Distribution Table  
X\Y  
0
1
0
1
0.3 0.2  
0.1 0.4  
This table gives P(X = x, Y = y) for each pair (x, y).  
 
Step 1: Marginal Probabilities  
Compute Marginal of X:  
P(X = 0) = 0.3 + 0.2 = 0.5  
P(X = 1) = 0.1 + 0.4 = 0.5  
Compute Marginal of Y :  
P(Y = 0) = 0.3 + 0.1 = 0.4  
P(Y = 1) = 0.2 + 0.4 = 0.6  
 
Step 2: Check for Independence  
Check if P(X = x, Y = y) = P(X = x) × P(Y = y) for all x, y.  
Example:  
P(X = 0, Y = 0) = 0.3  
P(X = 0) × P(Y = 0) = 0.5 × 0.4 = 0.2  
Since 0.3 ̸= 0.2, X and Y are NOT independent.  
 
Conclusion  
If P(X = x, Y = y) ̸= P(X = x)P(Y = y) for any (x, y), variables  
are dependent.  
Independence implies that knowing X tells us nothing about Y , and  
vice versa.  
Checking joint vs marginal probabilities is the key!  
 
Independence and Conditional Distributions  
Indpendence of Random Variabes  
Two random variables X and Y are independent if:  
P(X = x, Y = y) = P(X = x) × P(Y = y) for all x, y  
Interpretation: Knowing X does not affect beliefs about Y .  
il i
If independent:  
P(Y = y | X = x) = P(Y = y)  
(Conditioning on X does not change the distribution of Y .)  
If not independent:  
P(Y = y | X = x) ̸= P(Y = y)  
(Knowing X does change the distribution of Y .)  
 
Independence vs Dependence: A Comparison  
Property  
Joint Probability  
Conditional Probability  
Interpretation  
Independent  
P(X, Y ) = P(X)P(Y )  
P(Y |X) = P(Y )  
Dependent  
P(X, Y ) ̸= P(X)P(Y )  
P(Y |X) changes with X  
X and Y do not affect each other X and Y influence each other  
Key Point:  
Independence means the conditional distribution is the same as the  
marginal distribution.  
 
Population and Sample  
 
Population  
Population refers to the entire set of individuals, items, or data  
points that we are interested in studying.  
The population can be finite or infinite, and it contains all members  
from which we wish to draw conclusions.  
Key metrics for a population include:  
Total Population size (N) ; Size of the population.  
Population Mean (µ): The average of all the values in the  
population.  
Population Variance (σ2): Measures the spread or dispersion of the  
values in the population.  
Population Standard Deviation (σ): The square root of the  
population variance, representing the average distance of the data  
points from the population mean.  
 
Sample  
A sample is a subset of the population that is selected for the  
purpose of statistical analysis.  
It is often impractical or impossible to collect data from an entire  
population.  
so samples are used to make inferences about the population.  
Key metrics for a sample include:  
Total Sample size (n; Size of the sample.  
Sample Mean (x¯): The average of all the values in the sample.  
Sample Variance (s2): Measures the spread or dispersion of the  
values in the sample.  
Sample Standard Deviation (s): The square root of the sample  
variance, representing the average distance of the sample points from  
the sample mean.  
 
Comparison: Population vs. Sample  
The population and sample have similar metrics, but the key difference is  
that the population includes all members, while the sample is just a subset.  
Metric  
Mean  
Variance  
Population Sample  
µ
σ2  
σ
x¯  
s2  
s
Standard Deviation  
Proportion  
p
pˆ  
While the population parameters are fixed, the sample statistics are subject  
to sampling variability and serve as estimates of the population parameters.  
 
Population Parameter vs. Sample Statistic  
In statistics, we often distinguish between two key concepts:  
Population Parameter  
Sample Statistic  
These concepts are fundamental to understanding how we make inferences  
about a population based on a sample of data.  
 
Population Parameter  
A population parameter is a numerical value that summarizes a  
characteristic of an entire population.  
These parameters are fixed and unchanging.  
But they are often unknown because it’s usually impractical to  
measure an entire population.  
Examples of Population Parameters:  
Population Mean (µ)  
Population Variance (σ2)  
Population Standard Deviation (σ)  
Population Proportion (p)  
Key Point: Population parameters are fixed values, but they are often  
difficult or impossible to measure directly due to the size or accessibility of  
the population.  
 
Sample Statistic  
A sample statistic is a numerical value calculated from the data of a  
sample, which is a subset of the population.  
Sample statistics are used to estimate population parameters  
.
Examples of Sample Statistics:  
Sample Mean (x¯)  
Sample Variance (s2)  
Sample Standard Deviation (s)  
Sample Proportion (pˆ).  
Key Point: Sample statistics are variable, meaning they can change from  
sample to sample, even when taken from the same population.  
 
Key Differences: Population Parameter vs. Sample  
Statistic  
Aspect  
Definition  
Population Parameter  
A numerical value that summarizes a characteristic of the entire population.  
µ (mean), σ2 (variance), σ (standard deviation), p (proportion)  
Fixed and constant (if the population is known).  
Notation  
Fixed or Variable?  
Accessibility  
Usage  
Often unknown because it’s difficult to measure the entire population.  
Used to describe the entire population.  
Aspect  
Definition  
Sample Statistic  
A numerical value calculated from the data of a sample.  
x¯ (mean), s2 (variance), s (standard deviation), pˆ (proportion).  
Variable; changes depending on the sample.  
Notation  
Fixed or Variable?  
Accessibility  
Usage  
Known and easy to calculate because it’s based on the sample data.  
Used to estimate the corresponding population parameter.  
 
Inferential Statistics  
 
Sample Statistic: Its behaviour  
2
¯
A sample statistic (e.g., sample mean X, sample variance S ) is  
computed from a sample drawn from a population.  
Each time we draw a sample, the values in the sample may change.  
Since the sample changes randomly, the statistic computed from it  
also changes.  
Therefore, the statistic depends on random outcomes — it is a  
function of random variables.  
Conclusion: A sample statistic is a random variable because its  
value varies across different samples.  
It means Sample statistics will have a distribution.  
 
Inferential Statistics: Estimating Population Parameters  
Since it’s often impractical to measure an entire population, we rely on  
inferential statistics to make predictions or inferences about population  
parameters based on sample statistics.  
For example:  
If we want to know the average height of all high school students in a  
country, we might take a random sample of 500 students and  
calculate the sample mean height (x¯).  
The sample mean (x¯) can be used to estimate the population mean  
(µ).  
We can also compute confidence intervals and conduct hypothesis  
testing to make more precise inferences about the population  
parameter.  
Key Point:  
Sample statistics are used as estimates of population parameters.  
There is always some degree of uncertainty associated with these  
estimates.  
This can be quantified using confidence intervals, margin of error etc.  
 
Sampling and Inference  
Sample  
Sampling Process  
Population  
Sample  
Calculate  
Parameter Unknown  
Parameter  
Statistic  
Inference  
 
Research Question  
What is the average number of cups of coffee per day consumed by  
college students in the U.S.?  
Population: All college students in the U.S.  
Parameter of interest: Average daily coffee consumption  
Why We Sample  
It’s impractical to survey the entire population.  
We take a random sample to estimate the population mean.  
 
Example:  
Randomly select 30 students and ask how many cups of coffee they drink  
per day.  
Sample Data (Example) [2, 0, 1, 3, 2, 1, 2, 4, 0, 1, 2, 3,  
1, 2, 2, 1, 5, 1, 0, 3, 2, 1, 2, 0, 4, 3, 2, 1, 2, 1]  
Sample size: n = 30  
Sample mean: (computed visually)  
 
Each Response is a Random Variable  
Let Xi = cups of coffee consumed by the i-th student.  
Each Xi is a random variable:  
Depends on which student is randomly chosen  
Value varies with each sample  
¯
The sample mean X is also a random variable.  
Summary Table:  
Concept  
Example  
Population  
All U.S. college students  
Random Variable Xi Coffee cups consumed by student i  
Sample  
30 values like [2, 0, 1, ..., 1]  
Random selection  
Why It’s Random  
Selecting students at random  
Different students give different values  
 
Probability Distributions: Binomial  
Distributions  
 
Binomial Distribution  
A Bernoulli trial is an experiment or process that results in a binary  
outcome: success (usually coded as 1) or failure (usually coded as 0).  
The binomial distribution describes the number of successes in a  
fixed number of independent Bernoulli trials, each with the same  
probability of success.  
The binomial distribution can be defined by the following parameters:  
n : The number of trials.  
p : The probability of success on a single trial.  
k : The number of successes in n trials.  
The probability mass function (PMF) of the binomial distribution is  
given by:  
ꢀ ꢁ  
n
P(X = k) =  
pk(1 − p)n−k  
k
where:  
ꢂ ꢃ  
n
k
n!  
k!(n−k)!  
=
is the binomial coefficient.  
pk is the probability of having k successes.  
(1 − p)n−k is the probability of having n − k failures.  
 
Properties of the Binomial Distribution  
Some important properties of the binomial distribution:  
Mean: µ = n · p  
Variance: σ2 = n · p · (1 − p)  
p
Standard Deviation: σ = n · p · (1 − p)  
These properties help in understanding the central tendency and spread of  
the distribution.  
 
Binomial Distribution for different parameters  
 
Binomial Distribution for different parameters  
 
Example Problem  
Consider a scenario where a fair coin is flipped 10 times. What is the  
probability of getting exactly 6 heads?  
We know the following:  
n = 10 (number of flips),  
p = 0.5 (probability of getting a head),  
k = 6 (number of heads we are interested in).  
The probability mass function for the binomial distribution is:  
ꢀ
ꢁ
ꢀ
ꢁ
10  
6
10  
6
P(X = 6) =  
(0.5)6(0.5)10−6  
=
(0.5)10  
Now, calculating the binomial coefficient:  
ꢀ
ꢁ
10  
6
10!  
=
= 210  
6!4!  
Therefore, the probability is:  
1
P(X = 6) = 210 × (0.5)10 = 210 ×  
≈ 0.205  
1024  
 
Industrial Example: Quality Control in Manufacturing  
In a factory producing electronic components, the quality control  
department tests randomly selected parts for defects. Suppose the  
probability of a part being defective is 0.02 (2%).  
The company inspects 100 parts from a production batch. What is the  
probability that exactly 3 parts are defective?  
We have:  
n = 100 (number of parts inspected),  
p = 0.02 (probability of a part being defective),  
k = 3 (number of defective parts we are interested in).  
The probability mass function is:  
ꢀ
ꢁ
100  
3
P(X = 3) =  
(0.02)3(0.98)97  
First, calculate the binomial coefficient:  
ꢀ
ꢁ
100  
3
100!  
3!(100 − 3)!  
=
= 161700  
 
Example  
Now, calculate the probability:  
P(X = 3) = 161700 × (0.02)3 × (0.98)97  
This gives the probability of having exactly 3 defective parts in the  
batch.  
useful for manufacturers to predict and manage the likelihood of  
defective items and take corrective actions accordingly  
 
Binomial Distribution Functions in Excel  
Excel provides two main functions for the Binomial Distribution:  
BINOM.DIST(x, n, p, cumulative)  
x: Number of successes  
n: Number of trials  
p: Probability of success  
cumulative: TRUE for cumulative distribution function (CDF),  
FALSE for probability mass function (PMF)  
BINOM.DIST.RANGE(n, p, x1, [x2])  
Computes the probability of x1 to x2 successes (or exactly x1 if x2  
omitted)  
Example: =BINOM.DIST(3, 10, 0.5, FALSE) gives P(X = 3)  
Example: =BINOM.DIST.RANGE(10, 0.5, 3, 5) gives P(3 ≤ X ≤ 5)  
 
Excel Binomial Functions: Examples  
1. Using BINOM.DIST  
Question: A coin is flipped 8 times. What is the probability of getting  
exactly 5 heads?  
Solution: =BINOM.DIST(5, 8, 0.5, FALSE) ⇒ P(X = 5)  
2. Using BINOM.DIST.RANGE  
Question: A multiple-choice quiz has 10 questions, each with 4 options.  
What is the probability of guessing between 3 and 5 questions correctly?  
Solution: =BINOM.DIST.RANGE(10, 0.25, 3, 5) ⇒ P(3 ≤ X ≤ 5)  
 
BINOM.INV in Excel  
Function: BINOM.INV(trials, probability s, alpha)  
Returns the smallest value x such that:  
P(X ≤ x) ≥ α  
Useful for finding the inverse of the cumulative binomial distribution.  
Example Question: In a quality check of 20 items with a defect rate of  
10%, what is the smallest number of defective items such that the  
cumulative probability is at least 90%?  
Solution: =BINOM.INV(20, 0.1, 0.9) ⇒ Smallest x such that  
P(X ≤ x) ≥ 0.9  
 
Summary: Binomial Functions in Excel  
Function  
BINOM.DIST(x, n, p, FALSE)  
BINOM.DIST(x, n, p, TRUE)  
Purpose  
PMF  
Typical Use Case  
Probability of exactly x successes  
Probability of up to x successes  
CDF  
BINOM.DIST.RANGE(n, p, x1, [x2]) Range of PMF Probability that x1 ≤ X ≤ x2 (or exactly x1 if  
x2 omitted)  
BINOM.INV(n, p, α)  
Inverse CDF  
Smallest x such that P(X ≤ x) ≥ α  
Tip: Use BINOM.DIST.RANGE for probability intervals, and BINOM.INV to  
find thresholds based on cumulative probability.  
 
Probability Distributions: Poisson  
Distributions  
 
What is the Poisson Distribution?  
The Poisson distribution is a probability distribution that expresses the  
probability of a given number of events occurring in a fixed interval of time  
or space, given the average number of times the event occurs over that  
interval.  
It is typically used for rare events that happen independently of each other.  
Characteristics:  
The events are independent.  
The rate (λ) of occurrence is constant.  
The number of events in non-overlapping intervals are independent.  
The Poisson distribution is often used in situations like:  
Number of phone calls at a call center in an hour.  
Number of accidents at a traffic intersection in a day.  
Number of customers arriving at a store in a given time period.  
 
Poisson Distribution Formula  
The probability mass function (PMF) of the Poisson distribution is given  
by:  
λke−λ  
P(X = k) =  
,
k = 0, 1, 2, . . .  
k!  
where:  
X is the random variable representing the number of events.  
λ is the average number of events in the given time period (also  
called the rate parameter).  
k is the number of events observed.  
e is the Euler’s number, approximately 2.71828.  
This formula gives the probability of observing exactly k events when the  
expected number of events is λ.  
 
Mean and Variance of Poisson Distribution  
For a Poisson distribution with parameter λ:  
The mean is:  
µ = λ  
The variance is:  
σ2 = λ  
Notice that the mean and variance of a Poisson distribution are both equal  
to λ, which is an interesting property.  
 
Poisson Distribution for different parameters  
 
Example of Poisson Distribution  
Let’s say the average number of cars passing through a toll booth per hour  
is 5. We want to calculate the probability of exactly 3 cars passing  
through the booth in one hour.  
Here, the rate λ = 5 (the average number of cars per hour), and we want  
to find the probability when k = 3.  
Using the Poisson formula:  
53e−5  
P(X = 3) =  
3!  
125e−5  
P(X = 3) =  
6
P(X = 3) ≈ 0.1404  
So, the probability of exactly 3 cars passing through the toll booth in an  
hour is approximately 0.1404, or 14.04%.  
 
Applications of Poisson Distribution  
The Poisson distribution is widely used in various fields, such as:  
Telecommunications: Number of calls or messages in a given time  
frame.  
Healthcare: Number of patients arriving at an emergency room in a  
given hour.  
Traffic Flow: Number of cars passing a point in a road during a fixed  
period.  
Physics: Number of radioactive decay events in a time period.  
The Poisson distribution is particularly useful when events occur randomly  
and independently, and we are interested in counting how often they  
happen in a specific interval.  
 
Poisson Distribution Function in Excel  
Function: POISSON.DIST(x, mean, cumulative)  
x: Number of events  
mean: Expected number of events (λ)  
cumulative:  
TRUE – Cumulative distribution function P(X ≤ x)  
FALSE – Probability mass function P(X = x)  
Example: =POISSON.DIST(3, 4, FALSE) returns P(X = 3)  
Example: =POISSON.DIST(3, 4, TRUE) returns P(X ≤ 3)  
 
Poisson Function: Examples  
1. PMF with POISSON.DIST  
Question: A call center receives 4 calls per hour on average. What is the  
probability it receives exactly 3 calls in the next hour?  
Solution: =POISSON.DIST(3, 4, FALSE) ⇒ P(X = 3)  
2. CDF with POISSON.DIST  
Question: What is the probability the call center receives at most 3 calls?  
Solution: =POISSON.DIST(3, 4, TRUE) ⇒ P(X ≤ 3)  
 
Summary: Poisson Function in Excel  
Function  
POISSON.DIST(x, mean, FALSE) PMF  
POISSON.DIST(x, mean, TRUE) CDF  
Purpose Typical Use Case  
Probability of exactly x events in a fixed interval  
Probability of up to x events (cumulative)  
Tip: Use Poisson when modeling counts of rare events over time, space,  
or area.  
 
Probability Distributions: Normal  
Distributions  
 
What is the Normal Distribution?  
The Normal distribution is a continuous probability distribution that is  
symmetric about the mean. It is one of the most important distributions  
in statistics due to its natural occurrence in many real-world phenomena.  
Key Characteristics:  
It has a bell-shaped curve.  
The distribution is symmetric around the mean (µ).  
The mean, median, and mode of the distribution are all the same.  
The standard deviation (σ) controls the spread of the distribution.  
It is defined by two parameters: the mean (µ) and the standard  
deviation (σ).  
The Normal distribution is used to model many natural and social  
phenomena such as heights, test scores, and errors in measurements.  
 
Normal Distribution Formula  
The probability density function (PDF) of the Normal distribution is given  
by:  
2
(x−µ)  
1
e−  
2
2σ  
√
f(x) =  
σ 2π  
where:  
f(x) is the probability density at x.  
µ is the mean (average) of the distribution.  
σ is the standard deviation (a measure of the spread of the  
distribution).  
x is the variable of interest.  
e is Euler’s number (approximately 2.71828).  
The Normal distribution is fully described by its mean and standard  
deviation.  
 
Normal Distribution for different parameters  
 
Properties of the Normal Distribution  
Some key properties of the Normal distribution:  
Symmetry: The distribution is symmetric around the mean (µ).  
68-95-99.7 Rule:  
About 68% of the data falls within one standard deviation of the mean.  
About 95% falls within two standard deviations.  
About 99.7% falls within three standard deviations.  
Asymptotic: The tails of the Normal distribution approach, but  
never touch, the horizontal axis.  
Area under the Curve: The total area under the Normal curve is 1.  
 
Normal dist and standard deviation  
 
Applications of Normal Distribution  
The Normal distribution is used in a wide range of fields and applications:  
Psychology and Education: Test scores (e.g., IQ scores, SAT  
scores).  
Finance: Stock returns, asset prices.  
Healthcare: Measurements of physical traits like height, weight, and  
blood pressure.  
Manufacturing: Quality control, measurement of defects.  
Natural Sciences: Errors in scientific measurements, physical  
phenomena.  
The normal distribution is often used in hypothesis testing, confidence  
intervals, and regression analysis.  
 
Summary  
The Normal distribution is a continuous, symmetric distribution  
defined by its mean µ and standard deviation σ.  
It has a bell-shaped curve and is used to model many natural and  
social phenomena.  
The 68-95-99.7 Rule helps in understanding the spread of data in a  
Normal distribution.  
The Standard Normal distribution has µ = 0 and σ = 1, and Z-scores  
are used to standardize data.  
It is widely applied in statistics, finance, healthcare, and more.  
 
Normal Distribution  
 
Probability Distributions: Standard  
Normal Distributions  
 
What is the Standard Normal Distribution?  
A special case of the Normal distribution.  
It is a Normal distribution with a mean of 0 and a standard deviation  
of 1.  
Mathematically, the Standard Normal distribution is denoted by:  
Z ∼ N(0, 1)  
where:  
Z is the random variable.  
The mean (µ) is 0.  
The standard deviation (σ) is 1.  
The probability density function (PDF) of the Standard Normal  
distribution is:  
2
z
1
f(z) =  
e−  
2
√
2π  
where z is the standard score (also called the Z-score).  
 
Standard Normal Distribution  
Z-Score:  
x − µ  
z =  
σ
Applications:  
Standardization:  
Many datasets do not follow a Standard Normal distribution.  
Transform them into a Standard Normal form (using Z-scores).  
Z-Scores:  
Gives how many standard deviations a data point is from the mean.  
Used for identifying outliers and comparing data points from different  
distributions.  
Simplicity in Calculation:  
Well tabulated, simplifies calculations in hypothesis testing, confidence  
intervals, and many other statistical methods.  
Central Limit Theorem (CLT):  
The sampling distribution of the sample mean of a large enough sample  
from any population will be approximately normally distributed,  
regardless of the original population’s distribution.  
 
Z-Scores and Their Importance  
The Z-score is a key component of the Standard Normal distribution. It  
measures the relative position of a data point within a distribution. The  
Z-score is computed as:  
x − µ  
z =  
σ
where:  
x is the data point.  
µ is the mean of the distribution.  
σ is the standard deviation of the distribution.  
Interpretation of Z-scores:  
A Z-score of 0 means that the data point is exactly at the mean.  
A positive Z-score indicates the data point is above the mean.  
A negative Z-score indicates the data point is below the mean.  
Z-scores are essential for:  
Comparing scores from different normal distributions.  
Identifying outliers.  
 
Normal Distribution  
 
Observations from Z-Score Visualization  
Mean (µ) is shown as a black dot on each line and marked with a  
vertical dashed line for reference.  
Each data point is plotted on its own horizontal line, showing its  
distance from the mean.  
Z-scores are annotated beside each point, indicating how many  
standard deviations away the point is from the mean.  
Points within µ ± 2σ are marked in green, indicating they fall within  
the typical range of a normal distribution.  
Points beyond µ ± 2σ are marked in red, indicating they are  
statistical outliers.  
Vertical dashed lines mark standard deviation levels: µ ± 1σ, µ ± 2σ,  
and µ ± 3σ, helping to contextualize where each point lies.  
A point near 85 lies just beyond µ + 3σ, a rare event (occurs in less  
than 0.15% of data in a standard normal distribution).  
Similarly, points like 20 and 25 fall below µ − 2σ, also considered rare  
in a normal distribution.  
 
Standard Normal Distribution  
 
Observations from Standard Normal Z-Score Plot  
The mean (µ) of the standard normal distribution is at z = 0, marked  
with a vertical line.  
Each data point’s position is shown as a z-score, indicating how far it  
lies from the mean in units of standard deviation.  
Green points represent values within ±2σ, considered typical or  
common.  
Red points indicate data that lies beyond ±2σ, considered outliers in  
a standard normal distribution.  
Vertical dashed lines mark z = ±1, ±2, ±3, helping students  
understand how data is spread around the mean.  
The standard normal scale provides a universal reference to compare  
different datasets, regardless of their original units.  
The transformation helps visualize the relative rarity of extreme  
points.  
For example, the point at z > +3 is extremely rare, appearing in less  
than 0.15% of standard normal data.  
 
Comparison: Normal vs Standard Normal Representation  
Feature  
Normal  
Plot  
Distribution Standard Normal Plot  
X-Axis Scale  
Mean Location  
Original values (e.g., 45, Standardized z-scores (z)  
60, 85)  
Varies with data (e.g., Fixed at z = 0  
µ ≈ 50)  
Interpretation of Distance Contextual: depends on Universal: measured in σ  
unit scale  
Outliers Detection  
Based on absolute value Based on |z| > 2 or |z| >  
range  
3
Standard Deviation Lines  
Visualization Purpose  
µ ± 1σ, µ ± 2σ, etc.  
z = ±1, ±2, ±3  
Shows actual data spread Shows relative spread  
and units  
Based on deviation from Same (based on z mag-  
mean nitude)  
across datasets  
Color Coding  
 
Normal distribution curve  
 
Standard Normal distribution curve  
 
Standard Normal Table (Z-Table)  
The Standard Normal distribution has been extensively tabulated, with  
values for the cumulative distribution function (CDF) at different Z-scores.  
These values give the probability that a random variable is less than or  
equal to a given value in the Standard Normal distribution.  
For example, the Z-score table gives us:  
P(Z < 1.96) = 0.9750  
This means that the probability of a random variable being less than 1.96  
in the Standard Normal distribution is 97.5%.  
Z-tables are essential for:  
Finding probabilities for areas under the Normal curve.  
Hypothesis testing and confidence interval calculations.  
Determining critical values for statistical tests.  
 
Application of the Standard Normal Distribution  
The Standard Normal distribution is used in various areas of statistics and  
data analysis:  
Hypothesis Testing: In hypothesis testing, we standardize the test  
statistic (such as the Z-test) to compare it against the Standard  
Normal distribution.  
Confidence Intervals: The Z-distribution is used to calculate  
confidence intervals for population parameters (e.g., mean) when the  
population standard deviation is known.  
Central Limit Theorem (CLT): The Standard Normal distribution is  
a reference distribution in CLT for approximating the sampling  
distribution of the sample mean.  
Data Transformation: Converting a dataset into a Z-score  
(Standardizing) allows comparison across different datasets or  
variables.  
Error Detection: Z-scores are useful for identifying outliers in data,  
i.e., data points that are significantly higher or lower than the mean.  
 
Example:  
Let’s consider the heights of adult women in a population, which are  
normally distributed with a mean height of 64 inches and a standard  
deviation of 3 inches. We want to calculate the probability that a  
randomly selected woman has a height between 61 and 67 inches.  
First, we calculate the Z-scores for 61 inches and 67 inches:  
61 − 64  
67 − 64  
z1 =  
= −1 and z2 =  
= 1  
3
3
Now, we look up the cumulative probabilities for these Z-scores in the  
Standard Normal distribution table (or use a calculator). The probability  
for z1 = −1 is 0.1587 and for z2 = 1 is 0.8413.  
The probability that a woman’s height lies between 61 and 67 inches is:  
P(61 ≤ X ≤ 67) = P(z2) − P(z1) = 0.8413 − 0.1587 = 0.6826  
Thus, the probability is approximately 68.26%.  
 
Example: Standardizing a Dataset  
Suppose you have the following dataset representing the scores of students  
in two different classes:  
Class A: 70, 75, 80, 85, 90  
Class B: 55, 60, 65, 70, 75  
The means and standard deviations for each class are:  
µA = 80, σA = 7.9  
µB = 65, σB = 7.9  
 
Class A: Marks and Z-Scores  
Table: Class A (Mean = 80, SD =  
7.07)  
Table: Class B (Mean = 65, SD =  
7.07)  
Marks Z-Score  
Marks Z-Score  
70  
75  
80  
85  
90  
-1.41  
-0.71  
0
0.71  
1.41  
55  
60  
65  
70  
75  
-1.41  
-0.71  
0
0.71  
1.41  
The Z-scores are identical, indicating that a score of 85 in Class A and a  
score of 70 in Class B are equally above their respective means.  
 
Summary  
The Standard Normal distribution is a special case of the Normal  
distribution with mean 0 and standard deviation 1.  
It is essential for standardizing data and comparing different datasets  
with varying means and standard deviations.  
Z-scores represent the number of standard deviations a data point is  
from the mean and are used in hypothesis testing, confidence  
intervals, and error detection.  
Z-tables help compute cumulative probabilities and critical values for  
statistical tests.  
The Standard Normal distribution is widely used in fields like  
statistics, hypothesis testing, data analysis, and quality control.  
 
Summary: Normal Distribution Functions in Excel  
Function  
NORM.DIST(x, mean, sd, TRUE)  
Purpose  
CDF  
Typical Use Case  
Cumulative probability P(X ≤ x)  
Height of normal curve at x  
NORM.DIST(x, mean, sd, FALSE) PDF  
NORM.S.DIST(z, TRUE)  
NORM.S.DIST(z, FALSE)  
NORM.INV(prob, mean, sd)  
NORM.S.INV(prob)  
Standard Normal CDF P(Z ≤ z) from z-table  
Standard Normal PDF Height at z on standard normal curve  
Inverse CDF  
Inverse Std Normal  
Z-Score  
Returns x such that P(X ≤ x) = prob  
Returns z such that P(Z ≤ z) = prob  
x−µ  
STANDARDIZE(x, mean, sd)  
Converts raw score x to z =  
σ
Tip: Use STANDARDIZE to normalize values and apply standard normal  
functions easily.  
 
Probability Distributions:  
Chi-Square Distribution  
 
Introduction to the Chi-Square Distribution  
The Chi-Square distribution is a continuous probability distribution.  
It is used primarily in hypothesis testing, especially in goodness-of-fit  
tests.  
It is defined by the sum of the squares of independent standard  
normal random variables.  
 
Properties of the Chi-Square Distribution  
The Chi-Square distribution is parameterized by the degrees of  
freedom (df), which is typically denoted as k.  
The distribution is skewed to the right, especially for smaller degrees  
of freedom.  
Mean = k (where k is the degrees of freedom).  
Variance = 2k.  
 
Chi-Square Distribution Formula  
The probability density function (PDF) of the Chi-Square distribution is  
given by:  
x(k/2)−1e−x/2  
2k/2Γ(k/2)  
f(x; k) =  
,
x ≥ 0, k > 0  
Where:  
x is the random variable.  
k is the degrees of freedom.  
Γ is the Gamma function.  
 
Derivation of the Chi-Square Distribution  
The Chi-Square distribution can be derived from the normal distribution.  
Let Z1, Z2, . . . , Zk be independent standard normal variables. The  
Chi-Square random variable X is defined as:  
X = Z12 + Z22 + · · · + Zk2  
 
Derivation Steps  
Each Zi follows the standard normal distribution, i.e., Zi ∼ N(0, 1).  
The square of a standard normal variable Zi2 follows a Chi-Square  
distribution with 1 degree of freedom.  
The sum of squares of independent standard normal variables follows  
a Chi-Square distribution with k degrees of freedom.  
 
Example 1: Deriving Chi-Square for k = 2  
Let’s consider the case where k = 2, so we have two independent standard  
normal variables Z1 and Z2. Then the Chi-Square random variable X is:  
X = Z12 + Z22  
Each Zi ∼ N(0, 1), so we now have two independent variables squared  
and summed.  
The distribution of Z12 and Z22 is known as a Chi-Square distribution with  
1 degree of freedom.  
Thus, X = Z12 + Z22 ∼ χ22 (Chi-Square distribution with 2 degrees of  
freedom).  
 
Example 2: Deriving Chi-Square for k = 3  
Consider the case where k = 3. Now, we have three independent standard  
normal variables Z1, Z2, Z3. The Chi-Square random variable X is:  
X = Z12 + Z22 + Z32  
Each Zi ∼ N(0, 1), so the sum of their squares follows a Chi-Square  
distribution with 3 degrees of freedom. That is:  
X ∼ χ23  
This is a Chi-Square distribution with 3 degrees of freedom.  
 
Chi-Square Distribution with 2 and 3 Degrees of Freedom  
For k = 2:  
e−x/2  
f(x; 2) =  
,
x ≥ 0  
2
For k = 3:  
x(3/2)−1e−x/2  
23/2Γ(3/2)  
f(x; 3) =  
,
x ≥ 0  
We can observe that as the degrees of freedom increase, the distribution  
becomes more symmetric.  
 
Effect of Degrees of Freedom on the Shape of the  
Distribution  
The Chi-Square distribution is positively skewed when k is small.  
As the degrees of freedom increase, the distribution becomes more  
symmetric.  
For very large degrees of freedom, the distribution approximates a  
normal distribution.  
 
 
 
When to Use the Chi-Square Distribution  
The Chi-Square distribution is used in the following cases:  
Goodness-of-fit tests: To determine how well observed data fits a  
specific distribution (e.g., testing whether a die is fair).  
Test of independence: Used in contingency table analysis to test if  
two categorical variables are independent.  
Test for variance: In cases where we want to test if the variance of a  
population is equal to a specified value (e.g., testing if the variance of  
a sample matches the variance of the population).  
 
Chi-Square Distribution Functions in Excel  
1. CHISQ.DIST(x, deg freedom, cumulative)  
x – Test statistic  
deg freedom – Degrees of freedom  
cumulative – TRUE for CDF P(X ≤ x), FALSE for PDF  
2. CHISQ.INV(probability, deg freedom)  
Returns the value of x such that P(X ≤ x) = probability  
3. CHISQ.INV.RT(probability, deg freedom)  
Returns x such that P(X ≥ x) = probability — commonly used for  
hypothesis testing  
 
Summary: Chi-Square Functions in Excel  
Function  
Purpose  
CDF  
Typical Use Case  
Cumulative left-tail probability P(X ≤ x)  
Height of the chi-square curve at value x  
CHISQ.DIST(x, df, TRUE)  
CHISQ.DIST(x, df, FALSE)  
CHISQ.DIST.RT(x, df)  
PDF  
Right-tail probability Returns P(X ≥ x), used in right-tailed hypoth-  
esis tests  
CHISQ.INV(prob, df)  
Inverse CDF  
Returns x such that P(X ≤ x) = prob  
Returns x such that P(X ≥ x) = prob, for  
critical values  
CHISQ.INV.RT(prob, df)  
Inverse Right-Tail  
CHISQ.TEST(actual range, expected range) p-value from test  
Returns P(X ≥ x) for a test of independence  
or goodness-of-fit  
 
Probability Distributions: t-  
Distribution  
 
Introduction to t-Distributions  
The t-distribution is a family of probability distributions that arise  
when estimating population parameters from a small sample.  
It is used in hypothesis testing, particularly for small sample sizes.  
It is similar to the normal distribution but has heavier tails.  
The distribution is defined by a single parameter: the degrees of  
freedom (df).  
 
t-Distribution vs Normal Distribution  
The t-distribution approaches the standard normal distribution as the  
sample size increases.  
For large sample sizes (df > 30), the t-distribution and the normal  
distribution are nearly identical.  
The t-distribution has thicker tails, which allows for more variability in  
smaller samples.  
This characteristic helps account for the additional uncertainty in  
estimating the population mean from a small sample.  
 
Degrees of Freedom (df)  
Degrees of freedom (df) refer to the number of independent pieces of  
information used to estimate a statistical parameter.  
In the context of the t-distribution, df is typically n − 1, where n is  
the sample size.  
As df increases, the t-distribution approaches the normal distribution.  
 
Probability Density Function (PDF) of the t-Distribution  
The probability density function (PDF) of the t-distribution is given by:  
ꢂ
ꢃ
ν+1  
2
ꢀ
ꢁ−  
ν+1  
2
t2  
ν
Γ
ꢂ ꢃ  
f(t) =  
1 +  
√
ν
2
νπΓ  
Where:  
ν is the degrees of freedom,  
Γ is the Gamma function.  
 
 
Summary  
The t-distribution is used for small sample sizes and is particularly  
important in hypothesis testing.  
As the sample size increases, the t-distribution approaches the normal  
distribution.  
The degrees of freedom determine the shape of the t-distribution.  
The t-statistic is used to perform hypothesis tests, and the p-value  
helps make a decision.  
 
Summary: t-Distribution Functions in Excel  
Function  
Purpose  
Typical Use Case  
T.DIST(x, df, TRUE)  
T.DIST.RT(x, df)  
CDF (left-tail)  
Right-tail probability  
Returns P(T ≤ x)  
Returns P(T  
≥
x); common for one-tailed  
tests  
T.DIST.2T(x, df)  
Two-tail probability  
Returns P(|T| ≥ |x|); common for two-tailed  
tests  
T.INV(prob, df)  
Inverse CDF (left-tail) Returns x such that P(T ≤ x) = prob  
T.INV.2T(prob, df)  
Inverse two-tail  
Returns x such that P(|T| ≥ |x|) = prob  
Returns the p-value for a t-test between two  
samples  
T.TEST(array1, array2, tails, type) Hypothesis testing  
Tip: Use T.TEST to compare two means; use T.DIST.RT or T.DIST.2T if  
manually computing test statistics.  
 
Probability Distributions:F -  
Distribution  
 
Introduction to F-Distributions  
The F-distribution is a continuous probability distribution that arises  
from the ratio of two independent chi-squared variables.  
It is used primarily in hypothesis testing, especially for comparing  
variances in two populations.  
The F-distribution is not symmetric, and it only takes positive values.  
The distribution is determined by two degrees of freedom: one for the  
numerator and one for the denominator.  
Define the random variable F as the ratio of scaled chi-squared  
variables:  
X1/df1  
F =  
X2/df2  
 
F-Distribution vs Other Distributions  
The F-distribution is used in analysis of variance (ANOVA) and  
regression analysis.  
Unlike the normal distribution or t-distribution, the F-distribution is  
not symmetric and is right-skewed.  
It has two parameters, the degrees of freedom for the numerator (df1)  
and the denominator (df2).  
As df1 and df2 increase, the distribution becomes more symmetric.  
 
Degrees of Freedom (df) in F-Distribution  
The degrees of freedom are crucial in determining the shape of the  
F-distribution.  
The numerator degrees of freedom (df1) typically represent the  
number of groups or treatments.  
The denominator degrees of freedom (df2) often represent the number  
of observations within each group or the error degrees of freedom.  
The shape of the distribution depends on both df1 and df2. Larger  
degrees of freedom lead to a more symmetric distribution.  
 
F-Distribution  
The F-distribution is a continuous probability distribution.  
It arises in comparing two sample variances (e.g., ANOVA, regression).  
It is defined by two parameters: degrees of freedom:  
Fm,n where m = dfnumerator, n = dfdenominator  
The distribution is right-skewed and non-negative.  
A large F-statistic suggests a significant difference between variances  
or group means.  
 
Critical F-value and Alpha  
α is the significance level (e.g., 0.05).  
Understand α as area under the curve.  
It defines the probability of rejecting the null hypothesis when it’s true  
(Type I error).  
The critical value is denoted:  
Fα,m,n  
This value cuts off the right tail of the distribution such that:  
P(F > Fα,m,n) = α  
 
F-Distributions  
The red area under the curve represents α.  
Fα,m,n is the value such that the area to its right equals α.  
Prob of F3,20 < 3.10 = 1 − F0.05,3,20.  
 
Applications of the F-Distribution  
The F-distribution is widely used in several applications, including:  
Analysis of Variance (ANOVA): Used to compare the means of  
three or more groups. The F-test is used to determine if the group  
means are significantly different.  
Regression Analysis: The F-distribution is used to test the overall  
significance of a regression model, comparing the explained variance  
to the unexplained variance.  
Testing Variance Ratios: Used to test the hypothesis that two  
populations have equal variances.  
Model Comparisons: The F-distribution can be used to compare the  
fit of two competing models in statistics, such as nested models.  
 
F-Distribution Reciprocal Property  
The F-distribution arises when comparing two sample variances:  
s21  
s22  
F =  
∼ Fd ,d  
1
2
Used in ANOVA and hypothesis tests for comparing population  
variances.  
Typically, we use the right-tail of the distribution for significance  
tests.  
The F-distribution is asymmetric and right-skewed. However, it  
exhibits a special reciprocal relationship between the degrees of  
freedom.  
 
Reciprocal Identity of F-distributions  
Fundamental Property:  
1
F ∼ Fd ,d  
⇒
∼ Fd ,d  
1
2
2
1
F
Implication on tail probabilities:  
ꢀ
ꢁ
1
P(Fd ,d > f) = P Fd ,d  
<
1
2
2
1
f
A right-tail probability in one F-distribution becomes a left-tail  
probability in the reciprocal.  
Especially useful when test statistic is reversed.  
 
Summary  
The F-distribution is not symmetric — but it has a reciprocal  
symmetry:  
1
F ∼ Fd ,d  
⇒
∼ Fd ,d  
1
2
2
1
F
This allows us to:  
Transform right-tail areas into left-tail areas  
Understand the behavior of reversed F-ratios  
Simplify computations across software and tables  
 
Summary  
The F-distribution is used to compare variances between two or more  
populations.  
It is commonly applied in hypothesis testing (e.g., comparing  
variances, ANOVA).  
The distribution is determined by two degrees of freedom: numerator  
(df1) and denominator (df2).  
A large F-statistic suggests a significant difference between variances  
or group means.  
The F-distribution is derived from the ratio of two chi-squared  
distributions, and it plays a key role in regression analysis, model  
comparisons, and hypothesis testing.  
 
F-distribution  
 
Reciprocal F-distribution  
 
Summary: F-Distribution Functions in Excel  
Function  
Purpose  
Typical Use Case  
Returns P(F ≤ x); cumulative probability from  
the left  
F.DIST(x, df1, df2, TRUE) CDF (left-tail)  
F.DIST.RT(x, df1, df2)  
F.INV(prob, df1, df2)  
F.INV.RT(prob, df1, df2)  
Right-tail probability  
Returns P(F ≥ x); used in right-tailed F-tests  
Inverse CDF (left-tail) Returns x such that P(F ≤ x) = prob  
Inverse right-tail  
p-value for F-test  
Returns x such that P(F ≥ x) = prob; critical  
value for right-tailed test  
F.TEST(array1, array2)  
Returns the right-tailed p-value for comparing  
variances of two samples  
Tip: Use F.TEST for variance comparisons. Use F.INV.RT to find the  
critical value from a significance level.  
 
When to Use Each Distribution  
Binomial: Binary outcomes, fixed trials (e.g., coin flips, pass/fail  
tests).  
Poisson: Count of events in a fixed interval (e.g., arrivals per hour).  
Normal: Continuous data, central limit theorem applies (e.g., IQ  
scores).  
Chi-Square: Hypothesis testing for categorical data (e.g.,  
independence test).  
t-Distribution: Small sample means, unknown variance (e.g., student  
test scores).  
F-Distribution: Comparing variances or ANOVA (e.g., multiple  
group comparisons).  
 
Central Limit Theorem (CLT)  
 
Introduction  
The Central Limit Theorem (CLT) is a fundamental theorem in  
probability theory.  
It states that, under certain conditions, the sum (or average) of a  
large number of independent and identically distributed (i.i.d.)  
random variables follows a normal distribution, regardless of the  
original distribution.  
 
Formal Definition  
Let X1, X2, . . . , Xn be a sequence of i.i.d. random variables with mean µ  
and variance σ2. Define the sample mean as:  
n
X
1
¯
Xn =  
Xi.  
n
i=1  
Then, as n → ∞,  
¯
Xn − µ  
d
√
−→ N(0, 1).  
σ/ n  
This means the standardized sample mean converges in distribution to a  
standard normal distribution.  
 
Why is CLT Important?  
Enables statistical inference when population distribution is unknown.  
Justifies the normality assumption in many practical applications.  
Forms the basis for confidence intervals and hypothesis testing.  
 
Illustration: CLT in Action  
Consider different distributions (e.g., uniform, exponential, binomial).  
Take samples of increasing size and compute sample means.  
Observe how the distribution of sample means approaches a normal  
distribution.  
 
For 3000 sample means  
 
For sample size 30 and 1000 samples  
 
For sample size 1000 and 500 samples  
 
For Fixed sample size  
 
For Fixed sample size  
 
For Fixed sample size  
 
For any distribution  
 
For different sample sizes  
 
Example Application  
Suppose the heights of students in a university are not normally  
distributed. However:  
If we randomly select 30 students and compute the sample mean,  
The sample mean follows approximately a normal distribution.  
This allows us to make statistical inferences using normality-based  
methods.  
 
Limitations and Assumptions  
CLT requires a sufficiently large sample size (n ≥ 30 is a common rule  
of thumb).  
Assumes independence of random variables.  
Works best when individual observations have finite variance.  
 
CONFIDENCE INTERVALS  
 
Introduction to Confidence Intervals  
A confidence interval (CI) is a range of values, derived from sample  
statistics, that is likely to contain the true population parameter.  
It provides an estimate along with an indication of the estimate’s  
reliability.  
Confidence intervals are widely used in inferential statistics to make  
predictions about population parameters based on sample data.  
The confidence level represents the probability that the confidence  
interval contains the true population parameter.  
Common confidence levels: 90%, 95%, 99%.  
A higher confidence level means a wider interval, increasing the  
certainty of capturing the population parameter.  
Example: “We are 95% confident that the true mean lies between 50  
and 60.”  
 
Formula for Confidence Interval  
For a population mean µ with known standard deviation σ, the  
confidence interval is given by:  
σ
¯
√
X ± Zα/2  
n
where:  
¯
X = sample mean,  
Zα/2 = critical value from the standard normal distribution,  
σ = population standard deviation,  
n = sample size.  
 
Interpreting Confidence Intervals  
A 95% confidence interval means that if we were to take 100 different  
samples and compute a CI for each, about 95 of them would contain  
the true population parameter.  
It does not mean that there is a 95% probability that the true  
parameter is within the given interval.  
Effect of Sample Size on CI  
Larger sample sizes lead to narrower confidence intervals, increasing  
precision.  
Smaller sample sizes lead to wider confidence intervals, reflecting  
greater uncertainty.  
 
Effect of Confidence Level on CI  
Increasing the confidence level (e.g., from 95% to 99%) makes the  
interval wider to ensure greater certainty.  
Decreasing the confidence level (e.g., from 95% to 90%) makes the  
interval narrower but with less certainty.  
 
Conclusions from Different Confidence Intervals  
Wide Confidence Interval: Indicates more variability in data or  
smaller sample size.  
Narrow Confidence Interval: Suggests more precise estimates due  
to a larger sample size or lower variability.  
Overlapping Intervals: If two confidence intervals overlap, it  
suggests no significant difference between compared parameters.  
Non-Overlapping Intervals: Suggests a statistically significant  
difference between parameters.  
 
 
 
 
Confidence Intervals of Two Samples  
 
 
Real-World Applications  
Confidence intervals are used in medical research to determine the  
effectiveness of treatments.  
In finance, they help estimate stock market trends.  
In quality control, they assess manufacturing consistency.  
 
Limitations of Confidence Intervals  
Assumes random sampling; biased samples affect validity.  
Relies on the assumption that data follow a particular distribution.  
Wider intervals may not always provide useful conclusions.  
 
Conclusion  
Confidence intervals provide a way to estimate population parameters  
with a degree of certainty.  
The width of the interval depends on sample size and confidence level.  
Proper interpretation is crucial for making informed statistical  
decisions.  
 
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